The integral Hodge conjecture for two-dimensional Calabi–Yau categories. Issue 2 (12th February 2022)

• Record Type: Journal Article
• Authors: Perry, Alexander
• Is Part Of: Compositio mathematica. Volume 158:Issue 2(2022)
• Page Start: 287
• Page End: 333
• Publication Date: 2022-02-12
• Subjects: noncommutative variety -- integral Hodge conjecture -- Calabi–Yau category -- K3 surface -- intermediate Jacobian; 14F08 -- 14A22 -- 14C30 -- 14J28 -- 14J45; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X22007266
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi–Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use this to deduce cases of the usual integral Hodge conjecture for varieties. Along the way, we prove a version of the variational integral Hodge conjecture for families of two-dimensional Calabi–Yau categories, as well as a general smoothness result for relative moduli spaces of objects in such families. Our machinery also has applications to the structure of intermediate Jacobians, such as a criterion in terms of derived categories for when they split as a sum of Jacobians of curves.